Structures And Properties Of Some Infinite-dimensional Lie Algebras

Infinite-dimensional Lie algebras over a field of characteristic 0 have been intensively studied recently([27],[29-35]). In my thesis four classes of infinite dimensional Lie algebras over a field of characteristic 0(centerless Virasoro Lie algebra, Lie algebra L(Z, f,δ),subalgebras of rank 2 of Witt type Lie algebra and Necklace Lie algebra)are studied.Centerless Virasoro algebra, first appeared in 1909, was defined by E. Cartan. Yucai Su and Kaiming Zhao([27])prove that if g₁ is a finite-dimensional subalgebras of centerless Virasoro algebra, then dimg₁≤3, and if dimg₁=3, then exists a nonzero n∈Z such that g₁=Cd_n+Cd_O+Cd_-n. The problem of whether exist two dimensional Abelian Lie subalgebras in centerless Virasoro algebra mains open until now. By introducing the notion of coefficient matrix, we prove that Virasoro algebra does not have any 2-dimensional commutative subalgebra. We find some interesting 2-dimensional subalgebras of Virasoro algebra, apart from the obvious ones Cd_O+Cd_i, and we also discuss the properties of 2-dimensional subalgebras.It is a prime and interesting question how to determine the minimal number of generators and how to describe such the generators. M.Kuranishi proves that the minimal number of generators of finite-dimensional semi-simple Lie algebra of character 0 is two. Zhexian Wan and Caihui Lu([25-26])study the minimal number of generators of Kac-Moody Lie algebra g(A)and also describe the properties of the generators. In this paper, we prove the minimal number of generators of centerless Virasoro algebra is two. It is obtained that d_i and d_j are the minimal generators if and only if i and j axe relatively prime integers with different sign and with absolute value bigger than 1. The isomorphisms between subalgebras of the Virasoro algebra are discussed. The generic set, simplicity and maximal property of the subalgebras are also studied. In particular, We prove that g₁≌g₂ if and only if dim g₁=dim g₂.J. Marshall and Kaiming Zhao ([29])construct a class of infinite- dimensional Lie algebra L(A,δ,α) where A a torsion free Abelian group. They exhibit a large subclass of there algebras which are simple, as well as another subclass of these algebras which are never simple. The notion of transitive ideal plays an important role in this theory. They list two open problems.Question 1: Is Z(w)=Cw for all nonzero element w in simple Lie algebra L(A,δ,α)?Question 2: Determine Z(eo)in the Lie algebra L(A,δ,α).Caihui Lu and his doctor's student study the centralizer of the any element w in Lie algebra L(A,δ,α), and partly answer the two questions. Integer group Z takes place of a torsion free Abelian A in the third chapter and Lie algebra L(A,δ,α) is adapted Lie algebra L(Z, f,δ). Using a total ordering integer group Z and homomorphism's properties of integer group Z, we firstly define the notions of coefficient matrix and maximal element. We prove c(L(Z, f,δ)) = 0 and we prove that the Lie algebra is semi-simple and it has no two dimensional Abelian subalgebra. For the Lie algebra L(Z, f,δ), we answer the two open questions.Rank of 2 Lie algebra of Witt type is infinite-dimensional Lie algebra. Are the subalgebras simple Lie algebra in rank of 2 Lie algebra of Witt type ? We study the properties of its subalgebras (?)₁ and (?)₂ in the fourth chapter. We prove that c((?)₁) spanning by d₀-E₀ is Abelian and we also prove the linear space by d_k - E_k is Abelian ideal. We prove the subalgebra (?)₁ only has two Abelian ideal. We prove c((?)₂)=0 and Lie algebra (?)₂ isn't solvable and nilpotent. We prove that (?)₁ only have two Abelian ideals and (?)₂ only haven't Abelian ideals and (?)₁ is strong semi-simple algebra and (?)₂ is semi-simple algebra.Necklace Lie algebra is studied in the last chapter. Necklace Lie algebra induced by quiver graph is a new class of infinite-dimensional Lie algebra.We get some new results of Necklace Lie algebra, for instance, there are some finite-dimensional simple Lie subalgebras in Necklace Lie algebra, the finite-dimensional simple Lie subalgebras are isomorphic to simple sl(n). There are interesting properties of isomorphic and homomorphism of Necklace Lie algebra.